Contents

Idea

The kernel of a morphism is that part of its domain which is sent to 0.

Definition

There are various definitions of the notion of kernel, depending on the properties and structures available in the ambient category. We list a few definitions and discuss (in parts) when they are equivalent.

As a weighted limit

In any category enriched over pointed sets, the kernel of a morphism f:c→df:c\to d is the universal morphism k:a→ck:a\to c such that f∘kf \circ k is the basepoint. It is a weighted limit in the sense of enriched category theory. This applies in particular in any (pre)-additive category.

where kerker on the right-hand side is the kernel n the category of abelian groups.

If the category is in fact preabelian, kerfker f is also representable with representing object KerfKer f. One has to be careful with CokerfCoker f which does not represent the functor naive cokerfcoker f defined as (cokerf)(Z)=coker(hom(Z,X)→hom(Z,Y))(coker f)(Z) = coker(hom(Z,X)\to hom(Z,Y)) in AbAb, which is often not representable at all, even in the simple example of the category of abelian groups. Instead, as a colimit construction, one should corepresent another functor, namely, the covariant functor Z↦ker(hom(Y,Z)→hom(X,Z))Z\mapsto ker(hom(Y,Z) \to hom(X,Z)) (which is a quotient of the corepresentable functor hom(X,−)hom(X,-)). In short, CokerfCoker f is defined by the double dualization using the kernel in AbAb: Cokerf=(Kerfop)opCoker f = (Ker f^{op})^{op}. This is a particular case of the dualization involved in defining any colimit from its corresponding limit.

In an (∞,1)(\infty,1)-category

Other meanings

In some fields, the term ‘kernel’ refers to an equivalence relation that category theorists would see as a kernel pair. This is especially important in fields such as monoid theory where both notions exist but are not equivalent (while in group theory they are equivalent).

In ring theory, even when one assumes that rings have units preserved by ring homomorphisms, the traditional notion of kernel (an ideal) exists in the category of non-unital rings (and is not itself a unital ring in general). A purely category-theoretic theory of unital rings can be recovered either by using the kernel pair instead or (to fit better the usual language) moving to a category of modules.

Examples

Example

In the categoryAb of abelian groups, the kernel of a group homomorphismf:A→Bf : A \to B is the subgroup of AA on the set f−1(0)f^{-1}(0) of elements of AA that are sent to the zero-element of BB.

Example

More generally, for RR any ring, this is true in RRMod: the kernel of a morphism of modules is the preimage of the zero-element at the level of the underlying sets, equipped with the unique sub-module structure on that set.